An object (with charge -4.2 µC and mass 0.024 kg) hovers at rest above the ground, its weight being held up by a uniform electric field.(a) Find the size and direction of this electric field.(b) If the electric charge on the object is quadrupled while its mass remains the same, find the size and direction of its acceleration.
To solve these problems, we need to use the principles of electric forces and Newton's second law of motion. Let's break down the steps to find the answers to both parts of the question:
(a) Find the size and direction of the electric field:
Step 1: Calculate the weight of the object.
The weight (W) of an object is given by W = m * g, where m is the mass and g is the acceleration due to gravity (approximated as 9.8 m/s²).
Given:
Mass (m) = 0.024 kg
Using the formula:
W = m * g
W = 0.024 kg * 9.8 m/s²
W ≈ 0.2352 N
Step 2: Relate the weight to the electric force.
At equilibrium, the electric force on the object equals its weight. The electric force (Fe) is given by Fe = q * E, where q is the charge and E is the magnitude of the electric field.
Given:
Charge (q) = -4.2 µC (microCoulombs)
Weight (W) = 0.2352 N
Using the formula:
Fe = q * E
0.2352 N = (-4.2 × 10^-6 C) * E
Solving for E:
E = 0.2352 N / (-4.2 × 10^-6 C)
E ≈ -56,000 N/C
The negative sign indicates that the electric field direction is opposite to the direction of the weight force. So, the direction of the electric field is upward.
Therefore, the size of the electric field is approximately 56,000 N/C, and its direction is upwards.
(b) Find the size and direction of its acceleration:
When the charge on the object is quadrupled, the electric field will remain the same since it's determined by other factors. So, we'll use the initial electric field found in part (a).
Acceleration (a) is determined by the net force acting on the object divided by its mass (F = ma).
Given:
Charge (q) = 4 * (-4.2 µC) = -16.8 µC (microCoulombs)
Mass (m) = 0.024 kg
Using the formula:
Fe = q * E
Fe = (-16.8 × 10^-6 C) * (-56,000 N/C) = 0.9408 N
Now, rearrange the formula F = ma to find the acceleration:
F = ma
0.9408 N = (0.024 kg) * a
Solving for a:
a = 0.9408 N / 0.024 kg
a ≈ 39.2 m/s²
Therefore, the size of the acceleration is approximately 39.2 m/s², and its direction is the same as the electric field, which is upward.
(a) To find the size and direction of the electric field, we can use the equation that relates the electric field to the charge and weight of the object:
Electric field (E) = Weight (W) / Charge (q)
The weight of the object can be calculated using the formula:
Weight (W) = mass (m) * acceleration due to gravity (g)
Given:
Charge (q) = -4.2 µC = -4.2 x 10^-6 C (since 1 µC = 10^-6 C)
Mass (m) = 0.024 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the values into the formulas:
Weight (W) = 0.024 kg * 9.8 m/s^2 = 0.2352 N
Electric field (E) = 0.2352 N / (-4.2 x 10^-6 C) = -5.6 x 10^4 N/C
The size of the electric field is 5.6 x 10^4 N/C, and the direction is negative (in the opposite direction of the charge).
(b) If the charge on the object is quadrupled (4 times the original charge) while its mass remains the same, we need to find the new acceleration.
Using Newton's second law of motion, we have:
Force (F) = mass (m) * acceleration (a)
The electric force acting on the object can be calculated using the equation:
Electric force (F) = Charge (q) * electric field (E)
Given:
Charge (q) = 4 * -4.2 µC = -16.8 µC = -16.8 x 10^-6 C
Electric field (E) = -5.6 x 10^4 N/C
Mass (m) = 0.024 kg
Substituting the values into the formula:
Electric force (F) = (-16.8 x 10^-6 C) * (-5.6 x 10^4 N/C) = 0.9408 N
Now we can find the acceleration:
0.9408 N = 0.024 kg * acceleration (a)
Solving for acceleration (a):
acceleration (a) = 0.9408 N / 0.024 kg ≈ 39.2 m/s^2
The size of the acceleration is approximately 39.2 m/s^2, and it will be in the same direction as the electric field (negative direction).
Therefore, the size of the new acceleration is approximately 39.2 m/s^2, and its direction is negative (opposite to the electric field).